Why are the figer tips not moving with my hand in pose mode?
I belive I have recreated the problem with the image above
Check to see if the finger tips are weight painted. Weight painting controls how a model's mesh is manipulated whenever bones are moved. This means if a mesh is not painted correctly, you will get issues like part of the fingers not moving with the rest of the hand.
When in "Object" mode, select your armature, then also select your mesh which contains the fingertips

Weight painting mode 101:

    Single bone's weight paints can be viewed with ctrl and selecting that bone
    Multiple bone's weight paints can be viewed with shift and selcting multiple bones
    this mode will show a heatmap showing the weight of parts of the mesh
    
        dark blue = weight 0.0 -> this part of mesh is not affected when the selected bone is moved/ will               stay in place no matter how the bone(s) are moved
        dark red = weight 1.0 -> this part of mesh will completely move with the selected bone
    
    click on the mesh to paint on weights for certain bones

For more reading, here is the Blender Weight Painting Manual
So to fix the fingertips, you would select each of the bones at the tips of the fingers indivisually and paint the tops red with a brush with weight 1.0. This should make the fingertips snap into place as shown below. 

Once you've done this with each finger and its respective bone, you should be able to go into pose mode and pose it however you want without the issue

Hope this helps